Theorem isusp 24217

Description: The predicate 𝑊 is a uniform space. (Contributed by Thierry Arnoux, 4-Dec-2017.)

Hypotheses

Ref	Expression
isusp.1	⊢ 𝐵 = (Base‘𝑊)
isusp.2	⊢ 𝑈 = (UnifSt‘𝑊)
isusp.3	⊢ 𝐽 = (TopOpen‘𝑊)

Assertion

Ref	Expression
isusp	⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Proof of Theorem isusp

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3463	. 2 ⊢ (𝑊 ∈ UnifSp → 𝑊 ∈ V)
2		0nep0 5305	. . . . 5 ⊢ ∅ ≠ {∅}
3		isusp.1	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑊)
4		fvprc 6834	. . . . . . . . . . . 12 ⊢ (¬ 𝑊 ∈ V → (Base‘𝑊) = ∅)
5	3, 4	eqtrid 2784	. . . . . . . . . . 11 ⊢ (¬ 𝑊 ∈ V → 𝐵 = ∅)
6	5	fveq2d 6846	. . . . . . . . . 10 ⊢ (¬ 𝑊 ∈ V → (UnifOn‘𝐵) = (UnifOn‘∅))
7		ust0 24176	. . . . . . . . . 10 ⊢ (UnifOn‘∅) = {{∅}}
8	6, 7	eqtrdi 2788	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (UnifOn‘𝐵) = {{∅}})
9	8	eleq2d 2823	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 ∈ {{∅}}))
10		isusp.2	. . . . . . . . . 10 ⊢ 𝑈 = (UnifSt‘𝑊)
11	10	fvexi 6856	. . . . . . . . 9 ⊢ 𝑈 ∈ V
12	11	elsn 4597	. . . . . . . 8 ⊢ (𝑈 ∈ {{∅}} ↔ 𝑈 = {∅})
13	9, 12	bitrdi 287	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ 𝑈 = {∅}))
14		fvprc 6834	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (UnifSt‘𝑊) = ∅)
15	10, 14	eqtrid 2784	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝑈 = ∅)
16	15	eqeq1d 2739	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (𝑈 = {∅} ↔ ∅ = {∅}))
17	13, 16	bitrd 279	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ = {∅}))
18	17	necon3bbid 2970	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (¬ 𝑈 ∈ (UnifOn‘𝐵) ↔ ∅ ≠ {∅}))
19	2, 18	mpbiri 258	. . . 4 ⊢ (¬ 𝑊 ∈ V → ¬ 𝑈 ∈ (UnifOn‘𝐵))
20	19	con4i 114	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝐵) → 𝑊 ∈ V)
21	20	adantr 480	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)) → 𝑊 ∈ V)
22		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = (UnifSt‘𝑊))
23	22, 10	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifSt‘𝑤) = 𝑈)
24		fveq2 6842	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
25	24, 3	eqtr4di 2790	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝐵)
26	25	fveq2d 6846	. . . . 5 ⊢ (𝑤 = 𝑊 → (UnifOn‘(Base‘𝑤)) = (UnifOn‘𝐵))
27	23, 26	eleq12d 2831	. . . 4 ⊢ (𝑤 = 𝑊 → ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ↔ 𝑈 ∈ (UnifOn‘𝐵)))
28		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = (TopOpen‘𝑊))
29		isusp.3	. . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊)
30	28, 29	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = 𝑊 → (TopOpen‘𝑤) = 𝐽)
31	23	fveq2d 6846	. . . . 5 ⊢ (𝑤 = 𝑊 → (unifTop‘(UnifSt‘𝑤)) = (unifTop‘𝑈))
32	30, 31	eqeq12d 2753	. . . 4 ⊢ (𝑤 = 𝑊 → ((TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)) ↔ 𝐽 = (unifTop‘𝑈)))
33	27, 32	anbi12d 633	. . 3 ⊢ (𝑤 = 𝑊 → (((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤))) ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
34		df-usp 24213	. . 3 ⊢ UnifSp = {𝑤 ∣ ((UnifSt‘𝑤) ∈ (UnifOn‘(Base‘𝑤)) ∧ (TopOpen‘𝑤) = (unifTop‘(UnifSt‘𝑤)))}
35	33, 34	elab2g 3637	. 2 ⊢ (𝑊 ∈ V → (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈))))
36	1, 21, 35	pm5.21nii 378	1 ⊢ (𝑊 ∈ UnifSp ↔ (𝑈 ∈ (UnifOn‘𝐵) ∧ 𝐽 = (unifTop‘𝑈)))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442  ∅c0 4287  {csn 4582  ‘cfv 6500  Basecbs 17148  TopOpenctopn 17353  UnifOncust 24156  unifTopcutop 24186  UnifStcuss 24209  UnifSpcusp 24210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-res 5644  df-iota 6456  df-fun 6502  df-fv 6508  df-ust 24157  df-usp 24213

This theorem is referenced by:  ressust  24219  ressusp  24220  tususp  24227  uspreg  24229  ucncn  24240  neipcfilu  24251  ucnextcn  24259  xmsusp  24525